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Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints

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  • Touzi, Nizar

Abstract

We study the problem of minimal initial capital needed in order to hedge a European contingent claim without risk. The financial market presents incompleteness arising from two sources: stochastic volatility and portfolio constraints described by a closed convex set. In contrast with previous literature which uses the dual formulation of the problem, we use an original dynamic programming principle stated directly on the initial problem, as in Soner and Touzi (1998. SIAM J. Control Optim.; 1999. Preprint). We then recover all previous known results under weaker assumptions and without appealing to the dual formulation. We also prove a new characterization result of the value of super-replication as the unique continuous viscosity solution of the associated Hamilton-Jacobi-Bellman equation with a suitable terminal condition.

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  • Touzi, Nizar, 2000. "Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints," Stochastic Processes and their Applications, Elsevier, vol. 88(2), pages 305-328, August.
    • Handle: RePEc:eee:spapps:v:88:y:2000:i:2:p:305-328
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    References listed on IDEAS

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    1. Elyégs Jouini & Hédi Kallal, 1995. "Arbitrage In Securities Markets With Short‐Sales Constraints," Mathematical Finance, Wiley Blackwell, vol. 5(3), pages 197-232, July.
    2. Soner, H.M. & Touzi, N., 1999. "Stochastic Target Problems, Dynamic Programming and Viscosity Solutions," Papiers d'Economie Mathématique et Applications 1999.109, Université Panthéon-Sorbonne (Paris 1).
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    Cited by:

    1. Revaz Tevzadze & Teimuraz Toronjadze & Tamaz Uzunashvili, 2013. "Robust utility maximization for a diffusion market model with misspecified coefficients," Finance and Stochastics, Springer, vol. 17(3), pages 535-563, July.
    2. Bruno Bouchard & Ngoc-Minh Dang, 2013. "Generalized stochastic target problems for pricing and partial hedging under loss constraints—application in optimal book liquidation," Finance and Stochastics, Springer, vol. 17(1), pages 31-72, January.
    3. Josselin Garnier & Knut Solna, 2018. "Optimal hedging under fast-varying stochastic volatility," Papers 1810.08337, arXiv.org, revised Mar 2020.
    4. Bouchard, Bruno, 2002. "Stochastic targets with mixed diffusion processes and viscosity solutions," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 273-302, October.

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